Robust multi-way ANOVA, an approach based on the Forward Search

In this work, we present a simple robust method for the detection of atypical observations and the analysis of their effect in the ANalysis Of VAriance framework. In particular we will relate to the multi-way ANOVA, as an extension of the work by Bertaccini and Varriale (2007). The key statistic in the ANOVA is the F test of group means’ differences. The test is very powerful under classical assumptions, but it is strongly affected by the presence of outliers since it is based on the sample group mean that is not a robust statistic. Therefore, we propose to use a Forward Search procedure (Atkinson and Riani, 2000), a general powerful approach for detecting and investigating the effect of observations that differ from the bulk of the data on the inferential results of the model. The starting point is to fit an ANOVA model to very few observations chosen in a robust way, order all the observations by their closeness to the fitted model, increase the subset size, refit and continue until all the data are entered in the model. At every step of the search we compute parameters estimates, residuals, classical F values and all the other considerable statistics. The collection of these information is analysed (graphically or otherwise) in order to identify a cut-off point that divides the outliers from the other observations. The F value obtained at the cut-off point should be used to get a robust test. The goodness of the test will be evaluated simulating, through Monte Carlo studies, the real type I error probability and the test power. In particular, we are able to claim the clear superiority of our proposal since the probability of the type I error computed with the proposed method is much lower than the one computed with the classical approach when data are contaminated, without any loss in terms of power with “clean” data.